279
The effect of inadequate convergence criteria in automatic routines
By J. N. Lyness*
The possibility of round off or other statistical error in function values is sometimes not taken into
account in the construction of automatic routines. The possible consequences of such an omission
are discussed.
(Received February 1969)
'The aim of an automatic integration scheme is to
relieve the person who has to compute an integral-[the
user] of any need to think.' This is the opening sentence
of Chapter 6, 'Automatic Integration' of the standard
textbook Numerical Integration, by P. Davis and
P. Rabinowitz (1967).
Informal discussion indicates that views about this
aim vary from extreme to extreme. Some say the user
should be made to think; others, that it is impossible to
relieve him of all need to think and 'consequently' no
attempt should be made to relieve him of any need to
think. Another view is that this should be the aim of
all numerical analysis or at least the aim of any published
algorithm. Occasionally the same individual subscribes
to all of these views. However, the present author
accepts the opening sentence as a laudable aim, and
discusses below the presently available implementations
of this aim.
The textbook goes on to describe an automatic
integration scheme in general terms (as it appears to the
user). The user is to provide the upper and lower limits
B and A, a tolerance EP, a subroutine FUN(A') and N,
an upper limit on the number of function evaluations to
be used. After using N function evaluations the routine
gives up and provides the best result available and some
indication or message to this effect.
Although in the text it is suggested that N be provided
by the user, in the examples in the same book, N is
usually provided by the program. In the Romberg
Integration Code N = 215 + 1 = 32,769, and in the
Adaptive Simpson Code N = 2-3 30 + 1 ~ 4 X 1014.
Presumably experience has shown that it is difficult
enough to get the average user to think out what value
EP he requires. (This is the author's experience.)
When asked in addition for a value of TV, mental exhaustion
leads him to make some wild guess.
Thus in the present state of the art, the aim (which is
to relieve the user of any need to think) has been attained
to the following extent. He does not have to think (at
least not at this stage) about A, B, or FUN(JT). He is
forced to give some thought to EP. He has refused to
think about N and the programmer has usually done it
for him.
An automatic quadrature routine may be considered
as containing two essential ingredients. One is the
standard rule evaluation part, which is capable of
obtaining numerical approximations in terms of function
values. The other is the 'strategy component' using
which the routine considers the results it has already
obtained and decides what further calculation is necessary
to obtain a result of the desired accuracy.
An example is the automatic routine ROMBERG
{A, B, EP, FUN). The rule evaluation part is capable of
calculating successive approximations R{f R2f, •.., R\5f
to the required integral. Here Rjf is the standard
Romberg approximation requiring 2J+ 1 function values
(Bauer, Rutishauser, and Stiefel, 1963). The strategy
section is particularly simple. After each evaluation
Rjf, 2 > j > 14, the number
\Rjf-Rj-J\-EP
is calculated. If this number is less than or equal to
zero the routine returns Rjf as a final answer and the
calculation terminates. Otherwise the routine proceeds
to calculate the next approximation RJ+if
If j = 15,
the routine need not carry out this check. It simply
returns the result Rl5fa result requiring 32,769 function
evaluations.
In the Adaptive Simpson automatic quadrature routine, the procedure is more complicated. However, it is
still possible to distinguish between sections of coding
devoted to rule evaluation and sections devoted to
strategy.
We now turn to the question of the effect of round-off
error in these routines. While of course every single
operation in the routine could be considered as a source
of round-off error, the principal sources are generally
agreed to be these:
(i) Errors in the abcissas xt and weights w,-;
(ii) Errors in function values f(xt);
(iii) Accumulation Error in the sum
In general the effect of errors (i) and (iii) can be
removed by calculating abscissas and weights and carrying out the sum using double precision arithmetic. The
relative additional cost in computer time is not significant.
However, errors in the function values are beyond the
reach of the automatic routine.
From the point of view of the rule evaluation, such
errors while present are not serious. For example, if a
function is only accurate to seven digits the effect of
using a rule which is capable of obtaining a twelve digit
result is that a seven digit answer is obtained at a twelve
digit answer cost. Possibly a factor of 2 or 3 in machine
time might have been wasted. This is one reason why
round-off error is not considered a practical problem in
numerical quadrature.
However, from the point of view of strategy in an
automatic routine, moderate errors of a statistical nature
in the function values/(x,) may cause a disaster. Two
examples are illustrated in the accompanying tables.
* Argonne National Laboratory, Argonne, Illinois 60439
This work was performed under auspices of the U.S. Atomic Energy Commission.
280
J. N. Lyness
In each the same problem, the evaluation of
J
Table 2
Some results given by routine
cos x dx = — 1
o
is attempted to various accuracies EP, using essentially
the coding
FUN(Jf) = COS(X) + 1000000 0 - 1000000-0
(recoded in such a way that the compiler does not notice
the cancellation).
In the first example, Table 1, using ROMBERG
{A, B, EP, FUN) it is possible to see at a glance how the
strategy reacted to the information at its disposal; the
values \Rjf-Rj_J\
are listed. If EP = IO~6 the
routine concludes after stage j = 6 returning a six-figure
result based on 65 function evaluations. If EP = 10~7,
the routine goes on to stage./ = 1 5 returning a result of
comparable accuracy based on 32,769 function evaluations. The price paid by the user for not giving sufficient
thought to the value of EP is a factor 500 in machine
time, with no significant gain in accuracy.
Table 1
Romberg integration approximations
j
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
N
3
5
9
17
33
65
129
257
513
1,025
2,049
4,097
8,193
16,385
32,769
Rjf-Rj-if
-0-4879856539
-1-0127508941
-0-9998910218
—0-9999900407
-0-9999943948
-0-9999951011
-1-0000017132
-1-0000003520
-1-0000008370
-0-9999991700
-O-999999383O
- 1 0000001393
-0-9999999992
-1-0000001818
-0-9999997753
-5-25-001
1-29-002
-9-90-005
-4-35-006
-7-06-007
-6-61-006
1-36-006
-4-85-007
1-67-006
-2-13-007
-7-56-007
1-40-007
-1-83-007
4-07-007
In the second example, Table 2, using a modification
(Lyness, 1969) of the original version (McKeeman,
1962) of the adaptive Simpson Routine, the consequence
of asking for too much accuracy is even more catastrophic, a factor of 1,000 in machine time. In the table,
only results are given. Here if EP < 10~7 round-off
error prevents convergence in any interval unless the
difference to be tested is identically zero in that interval.
Essentially a machine accuracy calculation is carried out.
Depending on one's point of view, this type of
behaviour in these routines may be regarded as a fault,
or as a necessary penalty designed to influence the user
to give more thought to his problem. The fault, if it is
so considered, lies not in the rule evaluation section, but
in the strategy section. In general the numerical
approximations are all quite adequate. The simple
strategy in these routines assume that there will be no
significant round-off error and proceed on that basis. If
there is round-off error, the strategy sets the routine off
on a totally unrealistic sequence of calculations.
SQUARED (A, B, EP, FUN)
EP
N
io- 4
io-56
io-7
io-8
io-
29
45
97
152,997
152,997
ACTUAL ERROR*
5
-1-1 x io-6-9 x 10-6
3-9 x io- 6
-6-6 x io- 77
-6-6 x io-
• This is ( - 1 0-SQUARED) and not
(EXACT INTEGRAL-SQUARED)
In this example the round off level was set deliberately
to simulate a large cancellation error. Thus, if the
problem had been re-coded to avoid cancellation error
or if a computer using a longer word length had been
used, different results would have been obtained. This
is not the only way in which such errors arise. The
error in /(*,•) might be discretisation error arising from
perhaps a previous numerical quadrature. If an automatic routine is used for the previous calculation, the
discretisation error may have a different form for
different values of xt. Thus it would appear to the
automatic quadrature routine that the function f(x) it
is asked to integrate is smooth only in sections; the
errors here could be termed semi-statistical.
These errors may be relatively independent of the
machine accuracy. They stem from a desire on the part
of the user to economise at an earlier stage in the calculation. They are not due to round off nor are they due
to cancellation. Nevertheless they are present and may
cause the unwary routine to have difficulty in converging.
In general the function f(x) may be an intricate
combination of solutions of differential equations, spline
functions and even extrapolated experimental data. The
user may be unable or unwilling to perform an extensive
error analysis of the entire problem to determine the
accuracy of his function. Even if he were, to ask him
to do so is completely at variance with the stated aims
of automatic integration, namely that he should not have
to think at all. Instead of allowing him not to think,
these automatic routines demand that either he gives
considerable thought to the problem of determining EP,
or that he risks being penalised by computer time factors
of order 500 or 1,000.
The aims of automatic integration may be more
completely fulfilled if the strategy is adjusted to take into
account possible statistical error in the function values.
The intention should be that the routine, which has
available large numbers of function values /(*,), should
in some way use this information to determine the roundoff level—or 'noise'. The value of EP provided by the
user should be considered by the routine as a lower
bound only, and should be increased to the noise level
automatically if this becomes necessary. A user who
simply requires the best result available sets EP = 0 0.
At the end of the calculation EP is over-written by a
number calculated by the routine which represents its
own estimate of the accuracy attained.
The detailed method by which such a scheme should
Convergence criteria in automatic routines
be put into effect varies from routine to routine. The
author has implemented such a scheme in a modification
of the Adaptive Simpson routine described in detail in
Lyness (1969). This particular routine calculates in
any case various quantities which are very susceptible to
round-off error whose difference should generally be
positive; in practice this difference is used as an indicator
of the round-off level with reasonable success.
In principle a scheme of this nature can be implemented
by applying familiar theory concerned with the determination of the noise in a table of numbers. A chapter is
devoted to this in Hamming (1962). Or the criterion
could be simply that the routine terminates when successive iterates do not appear to be converging.
The author does not feel that there is any fundamental
difficulty in arranging for an automatic routine to take
into account the possibility of statistical fluctuations in
f(x,) and to terminate if appropriate. The principal
difficulty—and the main purpose of this article—is to
convince the people who construct such routines that it
is a necessary or worthwhile thing to do. Once people
are so convinced, they will be quite ingenious enough to
invent many schemes and to make thorough tests of
these with a view to producing the most efficient pro-
281
cedure for particular routines.
The discussion in this article is set entirely in the
context of Automatic Quadrature Routines. This is an
appropriate context. There is a general feeling that
round-off error in quadrature does not matter because
these errors do not become amplified but tend to eliminate each other. The point of this article that in any
automatic routine they may be critical is emphasised by
this context. The numerical stability of the calculations
is a separate question, which is of much less importance
in quadrature than in differentiation or the solution of
differential equations. Thus the foregoing discussion
applies to some extent to any Automatic routine which
requires a user-provided function subroutine.
Perhaps it should be emphasised that this sort of
procedure does not 'eliminate the round-off error'. The
final result is as good—or bad—as the given function,
the set of quadrature rules and the particular strategy
allow. All that this modification attempts to do is to
prevent round-off error (of a statistical nature) affecting
the strategy section in such a way that an extensive
sequence of unnecessary calculations is embarked upon
which do not improve the result, but which may use up
an excessive amount of computer time.
References
F. L., RUTISHAUSER, H., and STIEFEL, E. (1963). New aspects in numerical quadrature, Proc. Symposia Appl. Math
Vol. 15, pp. 199-218.
DAVIS, P., and RABINOWITZ, P. (1967). Numerical Integration, London: Blaisdell Publishing Co.
HAMMING, R. W. (1962). Numerical Methods for Scientists and Engineers, New York: McGraw-Hill Co., Inc.
LYNESS, J. N. (1969). Notes on the Adaptive Simpson Quadrature Routine, JACM, Vol. 16, No. 3.
MCKEEMAN, W. M. (1962). Algorithm 145, adaptive numerical integration by Simpson's Rule, Comm. ACM, Vol. 5, p. 604.
BAUER,